June  2021, 10(2): 229-247. doi: 10.3934/eect.2020063

Periodic solutions and multiharmonic expansions for the Westervelt equation

Department of Mathematics, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt, Austria

Received  August 2019 Revised  March 2020 Published  June 2021 Early access  June 2020

In this paper we consider nonlinear time periodic sound propagation according to the Westervelt equation, which is a classical model of nonlinear acoustics and a second order quasilinear strongly damped wave equation exhibiting potential degeneracy. We prove existence, uniqueness and regularity of solutions with time periodic forcing and time periodic initial-end conditions, on a bounded domain with absorbing boundary conditions. In order to mathematically recover the physical phenomenon of higher harmonics, we expand the solution as a superposition of contributions at frequencies that are multiples of a fundamental excitation frequency. This multiharmonic expansion is proven to converge, in appropriate function spaces, to the periodic solution in time domain.

Citation: Barbara Kaltenbacher. Periodic solutions and multiharmonic expansions for the Westervelt equation. Evolution Equations and Control Theory, 2021, 10 (2) : 229-247. doi: 10.3934/eect.2020063
References:
[1]

A. AnvariF. Forsberg and A. E. Samir, A primer on the physical principles of tissue harmonic imaging, RadioGraphics, 35 (2015), 1955-1964.  doi: 10.1148/rg.2015140338.

[2]

L. Bjørnø, Characterization of biological media by means of their non-linearity, Ultrasonics, 24 (1986), 254-259. 

[3]

H. Brèzis and L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math., 31 (1978), 1–30. doi: 10.1002/cpa.3160310102.

[4]

R. Brunnhuber and P. M. Jordan, On the reduction of Blackstock's model of thermoviscous compressible flow via Becker's assumption, International Journal of Non-Linear Mechanics, 78 (2016), 131-132.  doi: 10.1016/j.ijnonlinmec.2015.10.008.

[5]

R. BrunnhuberB. Kaltenbacher and P. Radu, Relaxation of regularity for the Westervelt equation by nonlinear damping with application in acoustic-acoustic and elastic-acoustic coupling, Evol. Equ. Control Theory, 3 (2014), 595-626.  doi: 10.3934/eect.2014.3.595.

[6] J. Burgers, A mathematical model illustrating the theory of turbulence, in Advances in Applied Mechanics, Academic Press, Inc., 1948. 
[7]

V. BurovI. GurinovichO. Rudenko and E. Tagunov, Reconstruction of the spatial distribution of the nonlinearity parameter and sound velocity in acoustic nonlinear tomography, Acoustical Physics, 40 (1994), 816-823. 

[8]

C. A. Cain, Ultrasonic reflection mode imaging of the nonlinear parameter B/A: I. A theoretical basis, The Journal of the Acoustical Society of America, 80 (1986), 28-32.  doi: 10.1109/ULTSYM.1985.198640.

[9]

A. Celik and M. Kyed, Nonlinear wave equation with damping: Periodic forcing and non-resonant solutions to the Kuznetsov equation, ZAMM Z. Angew. Math. Mech, 98 (2018), 412-430.  doi: 10.1002/zamm.201600280.

[10]

A. Celik and M. Kyed, Nonlinear acoustics: Blackstock-Crighton equations with a periodic forcing term, J. Math. Fluid Mech., 21 (2019), no. 3, Paper No. 45, 12 pp. doi: 10.1007/s00021-019-0451-4.

[11]

T. Christopher, Finite amplitude distortion-based inhomogeneous pulse echo ultrasonic imaging, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 44 (1997), 125-139.  doi: 10.1109/58.585208.

[12]

R. D. Fay, Plane sound waves of finite amplitude, J. Acoust. Soc. Amer., 3 (1931), 222-241.  doi: 10.1121/1.1915557.

[13]

A. S. Fokas and J. T. Stuart, The time periodic solution of the Burgers equation on the half-line and an application to steady streaming, J. Nonlinear Math. Phys., 12 (2006), 302-314.  doi: 10.2991/jnmp.2005.12.s1.24.

[14]

M. Fontes and O. Verdier, Time-periodic solutions of the Burgers equation, J. Math. Fluid Mech., 11 (2009), 303-323.  doi: 10.1007/s00021-007-0260-z.

[15]

E. Fubini, Anomalies in the propagation of acoustic waves at great amplitude, Alta Frequenza, 4 (1935), 530-581. 

[16]

D. Givoli, Non-reflecting boundary conditions, J. Comput. Phys., 94 (1991), 1-29.  doi: 10.1016/0021-9991(91)90135-8.

[17]

T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta Numerica, 8 (1999), 47-106.  doi: 10.1017/S0962492900002890.

[18]

N. IchidaT. Sato and M. Linzer, Imaging the nonlinear ultrasonic parameter of a medium, Ultrasonic Imaging, 5 (1983), 295-299.  doi: 10.1177/016173468300500401.

[19]

H. R. Jauslin, H. O. Kreiss, and J. Moser, On the forced Burgers equation with periodic boundary conditions, In Differential Equations: La Pietra 1996 (Florence), Amer. Math. Soc., Providence, RI, 1999,133–153. doi: 10.1090/pspum/065/1662751.

[20]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst., Ser. B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.

[21]

P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910–2009, Mechanics Research Communications, 73 (2016), 127-139.  doi: 10.1016/j.mechrescom.2016.02.014.

[22]

B. Kaltenbacher, Mathematics of Nonlinear Acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447.

[23]

B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation, Discrete Contin. Dyn. Syst., Ser. S, 2 (2009), 503-523.  doi: 10.3934/dcdss.2009.2.503.

[24]

B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, in Discrete Contin. Dyn. Syst. 2011, 8th AIMS Conference. Suppl. Vol. II, 2011,763–773.

[25]

B. Kaltenbacher and I. Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay, Math. Nachr., 285 (2012), 295-321.  doi: 10.1002/mana.201000007.

[26]

B. Kaltenbacher, I. Lasiecka and M. A. Pospieszalska, Wellposedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 1250035, 34 pp. doi: 10.1142/S0218202512500352.

[27]

B. Kaltenbacher and M. Thalhammer, Fundamental models in nonlinear acoustics part I. Analytical comparison, Math. Models Methods Appl. Sci., 28 (2018), 2403-2455.  doi: 10.1142/S0218202518500525.

[28]

N. N. Kochina, On periodic solutions of Burgers' equation, J. Appl. Math. Mech., 25 (1962), 1597-1607.  doi: 10.1016/0021-8928(62)90138-7.

[29]

P. Kokocki, Effect of resonance on the existence of periodic solutions for strongly damped wave equation, Nonlinear Anal., 125 (2015), 167-200.  doi: 10.1016/j.na.2015.05.012.

[30]

N. Krylová, Periodic solutions of hyperbolic partial differential equation with quadratic dissipative term, Czechoslovak Math. J., 20 (1970), 375-405. 

[31]

V. Kuznetsov, Equations of nonlinear acoustics, Soviet Physics-Acoustics, 16 (1971), 467-470. 

[32]

M. B. Lesser and R. Seebass, The structure of a weak shock wave undergoing reflexion from a wall, Journal of Fluid Mechanics, 31 (1968), 501-528.  doi: 10.1017/S0022112068000303.

[33]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271.  doi: 10.1007/s00245-011-9138-9.

[34]

S. Meyer and M. Wilke, Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces, Evol. Equ. Control Theory, 2 (2013), 365-378.  doi: 10.3934/eect.2013.2.365.

[35]

A. Parker, On the periodic solution of the Burgers equation: A unified approach, Proc. Roy. Soc. London Ser. A, 438 (1992), 113-132.  doi: 10.1098/rspa.1992.0096.

[36]

A. Rozanova, The Khokhlov-Zabolotskaya-Kuznetsov equation, C. R. Math. Acad. Sci. Paris, 344 (2007), 337-342.  doi: 10.1016/j.crma.2007.01.010.

[37]

I. Shevchenko and B. Kaltenbacher, Absorbing boundary conditions for nonlinear acoustics: The Westervelt equation, J. Comput. Phys., 302 (2015), 200-221.  doi: 10.1016/j.jcp.2015.08.051.

[38]

G. Simonett and M. Wilke, Well-posedness and longtime behavior for the Westervelt equation with absorbing boundary conditions of order zero, J. Evol. Equ., 17 (2017), 551-571.  doi: 10.1007/s00028-016-0361-3.

[39]

G. Teschl, Ordinary differential equations and dynamical systems, Graduate Studies in Mathematics, Vol. 140, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/140.

[40]

S. Tsynkov, Numerical solution of problems on unbounded domains. A review, Appl. Numer. Math., 27 (1998), 465-532.  doi: 10.1016/S0168-9274(98)00025-7.

[41]

F. VarrayO. BassetP. Tortoli and C. Cachard, Extensions of nonlinear B/A parameter imaging methods for echo mode, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 58 (2011), 1232-44.  doi: 10.1109/TUFFC.2011.1933.

[42]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustic Society of America, 35 (1963), 535-537.  doi: 10.1121/1.1918525.

[43]

E. A. Zabolotskaya and R. V. Khokhlov, Quasi-plane waves in the non-linear acoustics of confined beams, Soviet Physics-Acoustics, 15 (1969), 35-40. 

[44]

D. ZhangX. Chen and X. F. Gong, Acoustic nonlinearity parameter tomography for biological tissues via parametric array from a circular piston source–Theoretical analysis and computer simulations, The Journal of the Acoustical Society of America, 109 (2001), 1219-1225.  doi: 10.1121/1.1344160.

[45]

Dong ZhangXiufen Gong and Shigong Ye, Acoustic nonlinearity parameter tomography for biological specimens via measurements of the second harmonics, The Journal of the Acoustical Society of America, 99 (1996), 2397-2402.  doi: 10.1121/1.415427.

show all references

References:
[1]

A. AnvariF. Forsberg and A. E. Samir, A primer on the physical principles of tissue harmonic imaging, RadioGraphics, 35 (2015), 1955-1964.  doi: 10.1148/rg.2015140338.

[2]

L. Bjørnø, Characterization of biological media by means of their non-linearity, Ultrasonics, 24 (1986), 254-259. 

[3]

H. Brèzis and L. Nirenberg, Forced vibrations for a nonlinear wave equation, Comm. Pure Appl. Math., 31 (1978), 1–30. doi: 10.1002/cpa.3160310102.

[4]

R. Brunnhuber and P. M. Jordan, On the reduction of Blackstock's model of thermoviscous compressible flow via Becker's assumption, International Journal of Non-Linear Mechanics, 78 (2016), 131-132.  doi: 10.1016/j.ijnonlinmec.2015.10.008.

[5]

R. BrunnhuberB. Kaltenbacher and P. Radu, Relaxation of regularity for the Westervelt equation by nonlinear damping with application in acoustic-acoustic and elastic-acoustic coupling, Evol. Equ. Control Theory, 3 (2014), 595-626.  doi: 10.3934/eect.2014.3.595.

[6] J. Burgers, A mathematical model illustrating the theory of turbulence, in Advances in Applied Mechanics, Academic Press, Inc., 1948. 
[7]

V. BurovI. GurinovichO. Rudenko and E. Tagunov, Reconstruction of the spatial distribution of the nonlinearity parameter and sound velocity in acoustic nonlinear tomography, Acoustical Physics, 40 (1994), 816-823. 

[8]

C. A. Cain, Ultrasonic reflection mode imaging of the nonlinear parameter B/A: I. A theoretical basis, The Journal of the Acoustical Society of America, 80 (1986), 28-32.  doi: 10.1109/ULTSYM.1985.198640.

[9]

A. Celik and M. Kyed, Nonlinear wave equation with damping: Periodic forcing and non-resonant solutions to the Kuznetsov equation, ZAMM Z. Angew. Math. Mech, 98 (2018), 412-430.  doi: 10.1002/zamm.201600280.

[10]

A. Celik and M. Kyed, Nonlinear acoustics: Blackstock-Crighton equations with a periodic forcing term, J. Math. Fluid Mech., 21 (2019), no. 3, Paper No. 45, 12 pp. doi: 10.1007/s00021-019-0451-4.

[11]

T. Christopher, Finite amplitude distortion-based inhomogeneous pulse echo ultrasonic imaging, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 44 (1997), 125-139.  doi: 10.1109/58.585208.

[12]

R. D. Fay, Plane sound waves of finite amplitude, J. Acoust. Soc. Amer., 3 (1931), 222-241.  doi: 10.1121/1.1915557.

[13]

A. S. Fokas and J. T. Stuart, The time periodic solution of the Burgers equation on the half-line and an application to steady streaming, J. Nonlinear Math. Phys., 12 (2006), 302-314.  doi: 10.2991/jnmp.2005.12.s1.24.

[14]

M. Fontes and O. Verdier, Time-periodic solutions of the Burgers equation, J. Math. Fluid Mech., 11 (2009), 303-323.  doi: 10.1007/s00021-007-0260-z.

[15]

E. Fubini, Anomalies in the propagation of acoustic waves at great amplitude, Alta Frequenza, 4 (1935), 530-581. 

[16]

D. Givoli, Non-reflecting boundary conditions, J. Comput. Phys., 94 (1991), 1-29.  doi: 10.1016/0021-9991(91)90135-8.

[17]

T. Hagstrom, Radiation boundary conditions for the numerical simulation of waves, Acta Numerica, 8 (1999), 47-106.  doi: 10.1017/S0962492900002890.

[18]

N. IchidaT. Sato and M. Linzer, Imaging the nonlinear ultrasonic parameter of a medium, Ultrasonic Imaging, 5 (1983), 295-299.  doi: 10.1177/016173468300500401.

[19]

H. R. Jauslin, H. O. Kreiss, and J. Moser, On the forced Burgers equation with periodic boundary conditions, In Differential Equations: La Pietra 1996 (Florence), Amer. Math. Soc., Providence, RI, 1999,133–153. doi: 10.1090/pspum/065/1662751.

[20]

P. M. Jordan, Second-sound phenomena in inviscid, thermally relaxing gases, Discrete Contin. Dyn. Syst., Ser. B, 19 (2014), 2189-2205.  doi: 10.3934/dcdsb.2014.19.2189.

[21]

P. M. Jordan, A survey of weakly-nonlinear acoustic models: 1910–2009, Mechanics Research Communications, 73 (2016), 127-139.  doi: 10.1016/j.mechrescom.2016.02.014.

[22]

B. Kaltenbacher, Mathematics of Nonlinear Acoustics, Evol. Equ. Control Theory, 4 (2015), 447-491.  doi: 10.3934/eect.2015.4.447.

[23]

B. Kaltenbacher and I. Lasiecka, Global existence and exponential decay rates for the Westervelt equation, Discrete Contin. Dyn. Syst., Ser. S, 2 (2009), 503-523.  doi: 10.3934/dcdss.2009.2.503.

[24]

B. Kaltenbacher and I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, in Discrete Contin. Dyn. Syst. 2011, 8th AIMS Conference. Suppl. Vol. II, 2011,763–773.

[25]

B. Kaltenbacher and I. Lasiecka, An analysis of nonhomogeneous Kuznetsov's equation: Local and global well-posedness; exponential decay, Math. Nachr., 285 (2012), 295-321.  doi: 10.1002/mana.201000007.

[26]

B. Kaltenbacher, I. Lasiecka and M. A. Pospieszalska, Wellposedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 1250035, 34 pp. doi: 10.1142/S0218202512500352.

[27]

B. Kaltenbacher and M. Thalhammer, Fundamental models in nonlinear acoustics part I. Analytical comparison, Math. Models Methods Appl. Sci., 28 (2018), 2403-2455.  doi: 10.1142/S0218202518500525.

[28]

N. N. Kochina, On periodic solutions of Burgers' equation, J. Appl. Math. Mech., 25 (1962), 1597-1607.  doi: 10.1016/0021-8928(62)90138-7.

[29]

P. Kokocki, Effect of resonance on the existence of periodic solutions for strongly damped wave equation, Nonlinear Anal., 125 (2015), 167-200.  doi: 10.1016/j.na.2015.05.012.

[30]

N. Krylová, Periodic solutions of hyperbolic partial differential equation with quadratic dissipative term, Czechoslovak Math. J., 20 (1970), 375-405. 

[31]

V. Kuznetsov, Equations of nonlinear acoustics, Soviet Physics-Acoustics, 16 (1971), 467-470. 

[32]

M. B. Lesser and R. Seebass, The structure of a weak shock wave undergoing reflexion from a wall, Journal of Fluid Mechanics, 31 (1968), 501-528.  doi: 10.1017/S0022112068000303.

[33]

S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Appl. Math. Optim., 64 (2011), 257-271.  doi: 10.1007/s00245-011-9138-9.

[34]

S. Meyer and M. Wilke, Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces, Evol. Equ. Control Theory, 2 (2013), 365-378.  doi: 10.3934/eect.2013.2.365.

[35]

A. Parker, On the periodic solution of the Burgers equation: A unified approach, Proc. Roy. Soc. London Ser. A, 438 (1992), 113-132.  doi: 10.1098/rspa.1992.0096.

[36]

A. Rozanova, The Khokhlov-Zabolotskaya-Kuznetsov equation, C. R. Math. Acad. Sci. Paris, 344 (2007), 337-342.  doi: 10.1016/j.crma.2007.01.010.

[37]

I. Shevchenko and B. Kaltenbacher, Absorbing boundary conditions for nonlinear acoustics: The Westervelt equation, J. Comput. Phys., 302 (2015), 200-221.  doi: 10.1016/j.jcp.2015.08.051.

[38]

G. Simonett and M. Wilke, Well-posedness and longtime behavior for the Westervelt equation with absorbing boundary conditions of order zero, J. Evol. Equ., 17 (2017), 551-571.  doi: 10.1007/s00028-016-0361-3.

[39]

G. Teschl, Ordinary differential equations and dynamical systems, Graduate Studies in Mathematics, Vol. 140, American Mathematical Society, Providence, RI, 2012. doi: 10.1090/gsm/140.

[40]

S. Tsynkov, Numerical solution of problems on unbounded domains. A review, Appl. Numer. Math., 27 (1998), 465-532.  doi: 10.1016/S0168-9274(98)00025-7.

[41]

F. VarrayO. BassetP. Tortoli and C. Cachard, Extensions of nonlinear B/A parameter imaging methods for echo mode, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 58 (2011), 1232-44.  doi: 10.1109/TUFFC.2011.1933.

[42]

P. J. Westervelt, Parametric acoustic array, The Journal of the Acoustic Society of America, 35 (1963), 535-537.  doi: 10.1121/1.1918525.

[43]

E. A. Zabolotskaya and R. V. Khokhlov, Quasi-plane waves in the non-linear acoustics of confined beams, Soviet Physics-Acoustics, 15 (1969), 35-40. 

[44]

D. ZhangX. Chen and X. F. Gong, Acoustic nonlinearity parameter tomography for biological tissues via parametric array from a circular piston source–Theoretical analysis and computer simulations, The Journal of the Acoustical Society of America, 109 (2001), 1219-1225.  doi: 10.1121/1.1344160.

[45]

Dong ZhangXiufen Gong and Shigong Ye, Acoustic nonlinearity parameter tomography for biological specimens via measurements of the second harmonics, The Journal of the Acoustical Society of America, 99 (1996), 2397-2402.  doi: 10.1121/1.415427.

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